Integrand size = 24, antiderivative size = 94 \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=\frac {x}{a}+\frac {b \text {arctanh}\left (\frac {b+2 c f^{g+h x}}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} h \log (f)}-\frac {\log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)} \]
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Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2320, 719, 29, 648, 632, 212, 642} \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c f^{g+h x}}{\sqrt {b^2-4 a c}}\right )}{a h \log (f) \sqrt {b^2-4 a c}}-\frac {\log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}+\frac {x}{a} \]
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Rule 29
Rule 212
Rule 632
Rule 642
Rule 648
Rule 719
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )} \, dx,x,f^{g+h x}\right )}{h \log (f)} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,f^{g+h x}\right )}{a h \log (f)}+\frac {\text {Subst}\left (\int \frac {-b-c x}{a+b x+c x^2} \, dx,x,f^{g+h x}\right )}{a h \log (f)} \\ & = \frac {x}{a}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,f^{g+h x}\right )}{2 a h \log (f)}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,f^{g+h x}\right )}{2 a h \log (f)} \\ & = \frac {x}{a}-\frac {\log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c f^{g+h x}\right )}{a h \log (f)} \\ & = \frac {x}{a}+\frac {b \tanh ^{-1}\left (\frac {b+2 c f^{g+h x}}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} h \log (f)}-\frac {\log \left (a+b f^{g+h x}+c f^{2 g+2 h x}\right )}{2 a h \log (f)} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98 \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=-\frac {\frac {2 b \arctan \left (\frac {b+2 c f^{g+h x}}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 \log \left (f^{g+h x}\right )+\log \left (a+f^{g+h x} \left (b+c f^{g+h x}\right )\right )}{2 a h \log (f)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(545\) vs. \(2(88)=176\).
Time = 0.10 (sec) , antiderivative size = 546, normalized size of antiderivative = 5.81
method | result | size |
risch | \(\frac {4 \ln \left (f \right )^{2} a c \,h^{2} x}{4 \ln \left (f \right )^{2} a^{2} c \,h^{2}-\ln \left (f \right )^{2} a \,b^{2} h^{2}}-\frac {\ln \left (f \right )^{2} b^{2} h^{2} x}{4 \ln \left (f \right )^{2} a^{2} c \,h^{2}-\ln \left (f \right )^{2} a \,b^{2} h^{2}}+\frac {4 \ln \left (f \right )^{2} a c g h}{4 \ln \left (f \right )^{2} a^{2} c \,h^{2}-\ln \left (f \right )^{2} a \,b^{2} h^{2}}-\frac {\ln \left (f \right )^{2} b^{2} g h}{4 \ln \left (f \right )^{2} a^{2} c \,h^{2}-\ln \left (f \right )^{2} a \,b^{2} h^{2}}-\frac {2 \ln \left (f^{h x +g}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 c b}\right ) c}{\left (4 c a -b^{2}\right ) h \ln \left (f \right )}+\frac {\ln \left (f^{h x +g}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 c b}\right ) b^{2}}{2 a \left (4 c a -b^{2}\right ) h \ln \left (f \right )}+\frac {\ln \left (f^{h x +g}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 c b}\right ) \sqrt {-4 a \,b^{2} c +b^{4}}}{2 a \left (4 c a -b^{2}\right ) h \ln \left (f \right )}-\frac {2 \ln \left (f^{h x +g}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 c b}\right ) c}{\left (4 c a -b^{2}\right ) h \ln \left (f \right )}+\frac {\ln \left (f^{h x +g}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 c b}\right ) b^{2}}{2 a \left (4 c a -b^{2}\right ) h \ln \left (f \right )}-\frac {\ln \left (f^{h x +g}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 c b}\right ) \sqrt {-4 a \,b^{2} c +b^{4}}}{2 a \left (4 c a -b^{2}\right ) h \ln \left (f \right )}\) | \(546\) |
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Time = 0.32 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.29 \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=\left [\frac {2 \, {\left (b^{2} - 4 \, a c\right )} h x \log \left (f\right ) + \sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} f^{2 \, h x + 2 \, g} + b^{2} - 2 \, a c + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} f^{h x + g} + \sqrt {b^{2} - 4 \, a c} b}{c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} h \log \left (f\right )}, \frac {2 \, {\left (b^{2} - 4 \, a c\right )} h x \log \left (f\right ) + 2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {2 \, \sqrt {-b^{2} + 4 \, a c} c f^{h x + g} + \sqrt {-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c f^{2 \, h x + 2 \, g} + b f^{h x + g} + a\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} h \log \left (f\right )}\right ] \]
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Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11 \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=\operatorname {RootSum} {\left (z^{2} \cdot \left (4 a^{2} c h^{2} \log {\left (f \right )}^{2} - a b^{2} h^{2} \log {\left (f \right )}^{2}\right ) + z \left (4 a c h \log {\left (f \right )} - b^{2} h \log {\left (f \right )}\right ) + c, \left ( i \mapsto i \log {\left (f^{g + h x} + \frac {- 4 i a^{2} c h \log {\left (f \right )} + i a b^{2} h \log {\left (f \right )} - 2 a c + b^{2}}{b c} \right )} \right )\right )} + \frac {x}{a} \]
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Exception generated. \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.17 \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=-\frac {\frac {2 \, b \arctan \left (\frac {2 \, c f^{h x} f^{g} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a \log \left (f\right )} + \frac {\log \left (c f^{2 \, h x} f^{2 \, g} + b f^{h x} f^{g} + a\right )}{a \log \left (f\right )} - \frac {2 \, \log \left ({\left | f \right |}^{h x} {\left | f \right |}^{g}\right )}{a \log \left (f\right )}}{2 \, h} \]
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Time = 0.49 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \frac {1}{a+b f^{g+h x}+c f^{2 (g+h x)}} \, dx=\frac {x}{a}-\frac {\ln \left (a+c\,f^{2\,h\,x}\,f^{2\,g}+b\,f^{h\,x}\,f^g\right )}{2\,a\,h\,\ln \left (f\right )}-\frac {b\,\mathrm {atan}\left (\frac {b+2\,c\,f^{h\,x}\,f^g}{\sqrt {4\,a\,c-b^2}}\right )}{a\,h\,\ln \left (f\right )\,\sqrt {4\,a\,c-b^2}} \]
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